Lesson 3: 9.3 Tests About a Difference of Proportions

1. How do we state a null hypothesis when we are comparing two groups (e.g.,H0: p1 - p2 = 0)?

2. Does a significant difference in proportions prove a cause-and-effect relationship?

AP Stats CED: SRE-2.D (Conditions for p1-p2), SRE-2.E (Calculating z and p-value for p1-p2), SRE-2.F (Conclusions). Common Core: HSS-IC.B.5.

Description
This section covers the Two-Sample z-Test for p1 - p2. The "Big Idea" here is the pooled proportion, where we combine the successes and sample sizes from both groups to get a single estimate of the proportion. This is used only when the null hypothesis assumes the two proportions are equal.

Purpose
To allow for comparative experiments and observational studies (e.g., "Is the graduation rate higher for Group A than Group B?"). It is the gold standard for testing the effectiveness of a treatment vs. a control in categorical data.

DOK Level
Level 3 (Strategic Thinking): Students must decide whether the data comes from a randomized experiment (allowing for causation) or an observational study (only allowing for association) and justify their conclusion based on the P-value.

Struggling Learners: Focus on the "Pooling" visual. Draw two jars of marbles. Group 1 has 10/50 red; Group 2 has 20/50 red. If we assume they are the same (the Null), we should just dump them both into one big jar (30/100) to find the "best" version of the truth.

Advanced Learners: Ask them to explain why we use the pooled proportion for a significance test (where we assume p1=p2) but we do not pool for a confidence interval (where we don't assume they are equal). This is a high-level conceptual distinction often tested on the AP Exam.

ELL Learners: Use a "Comparison Organizer" to list the "Successes" and "Total" for both groups clearly. Use the mathematical symbol not equal to represent "different," > for "more," and < for "less" to help them translate word problems into alternative hypotheses.

Applicatio activity; mastery-based assignments; Unit Test